Guessing, if there is a Heart Attack or not

Statistical Methods and Laboratory, Part II - The Final Project

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• The Introduction
  • The Dataset
  • The Data Visualization
    • The visualization using PCA
    • The Categorical Data
    • The Quantitative Data
    • The Correlations
• Preliminary brief definitions
• The Goal
  • The First Model (link function with logit)
    • Implementing RJags
    • The stationary regions
    • Some other interesting plots
    • The Cumulative Means
    • The Approximation Error
    • Posterior uncertainty
    • The Estimated Correlations
  • Hypotheses and Testing of the convergences of these multiple Markov chains
    • Raftery & Lewis Test
    • Geweke Test
    • Gelman Test
    • Heidel Test
  • Credible interval and the point estimates
  • The Predictions
• The Second Model
  • The comparison between these two models (DIC)
• The Conclusions
  • Simulation for parameters recovery
  • …Final Conclusions
  • The References

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The Introduction

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Nowadays, with the improvement of the technologies there are an amount of data that allows us to understand if there is any problem or not in the healthy of a person.

Here in this project we explain and we try to predict whether a patient should be diagnosed with Heart Disease or not.

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The Dataset

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Kaggle, is the main platform where I found this interesting dataset where applying our main bayesian inference as the main scope of this project.

This dataset is composed by these features:

## 
## -- Column specification --------------------------------------------------------
## cols(
##   age = col_double(),
##   sex = col_double(),
##   cp = col_double(),
##   trtbps = col_double(),
##   chol = col_double(),
##   fbs = col_double(),
##   restecg = col_double(),
##   thalachh = col_double(),
##   exng = col_double(),
##   oldpeak = col_double(),
##   slp = col_double(),
##   caa = col_double(),
##   thall = col_double(),
##   output = col_double()
## )

## # A tibble: 6 x 14
##     age   sex    cp trtbps  chol   fbs restecg thalachh  exng oldpeak   slp
##   <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl>   <dbl>    <dbl> <dbl>   <dbl> <dbl>
## 1    63     1     3    145   233     1       0      150     0     2.3     0
## 2    37     1     2    130   250     0       1      187     0     3.5     0
## 3    41     0     1    130   204     0       0      172     0     1.4     2
## 4    56     1     1    120   236     0       1      178     0     0.8     2
## 5    57     0     0    120   354     0       1      163     1     0.6     2
## 6    57     1     0    140   192     0       1      148     0     0.4     1
## # ... with 3 more variables: caa <dbl>, thall <dbl>, output <dbl>

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Analyzing these features, I recognized that the:

1. Age -> it refers to the age of the patient
2. sex -> it refers if the patient is:
   • male (1)
   • female (0)
3. cp -> it refers to the chest pain type that could be of four types:
   • 0 is the typical angina
   • 1 is the atypical angina
   • 2 is the non-anginal pain
   • 3 is the asymptomatic type
4. trestbps -> it refers to the resting blood pressure
5. chol -> it refers to the serum cholesterol in mg/dl
6. fbs -> it refers to the fasting blood sugar that if it is larger than 120 mg/dl is represented with:
   • 1 (true)
   • 0 (false)
7. restecg -> it refers to the resting electrocardiographic results:
   • 0 as Normal
   • 1 having ST-T wave abnormality (T wave inversions and/or ST elevation or depression of > 0.05 mV)
   • 2 showing probable or definite left ventricular hypertrophy by Estes’ criteria
8. thalachh -> it refers to the maximum heart rate achieved
9. exng -> it refers to the exercise induced angina that could be:
   • 1 (yes)
   • 0 (no)
10. oldpeak -> it refers to the Previous peak
11. slp -> it refers to the peak exercise ST segment:
    • 0 as up sloping
    • 1 as flat
    • 2 as down sloping
12. caa -> it refers to the number of major vessels that goes from 0 to 3
13. thall -> it refers to the diagnostic maximum heart rate achieved:
    • 0 as no-data
    • 1 as normal
    • 2 as fixed defect
    • 3 as reversible defect
14. output -> it refers to the fact to have the heart attack or not

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The main features detected are:

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##       age             sex               cp             trtbps     
##  Min.   :29.00   Min.   :0.0000   Min.   :0.0000   Min.   : 94.0  
##  1st Qu.:48.00   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:120.0  
##  Median :55.50   Median :1.0000   Median :1.0000   Median :130.0  
##  Mean   :54.42   Mean   :0.6821   Mean   :0.9636   Mean   :131.6  
##  3rd Qu.:61.00   3rd Qu.:1.0000   3rd Qu.:2.0000   3rd Qu.:140.0  
##  Max.   :77.00   Max.   :1.0000   Max.   :3.0000   Max.   :200.0  
##       chol            fbs           restecg          thalachh    
##  Min.   :126.0   Min.   :0.000   Min.   :0.0000   Min.   : 71.0  
##  1st Qu.:211.0   1st Qu.:0.000   1st Qu.:0.0000   1st Qu.:133.2  
##  Median :240.5   Median :0.000   Median :1.0000   Median :152.5  
##  Mean   :246.5   Mean   :0.149   Mean   :0.5265   Mean   :149.6  
##  3rd Qu.:274.8   3rd Qu.:0.000   3rd Qu.:1.0000   3rd Qu.:166.0  
##  Max.   :564.0   Max.   :1.000   Max.   :2.0000   Max.   :202.0  
##       exng           oldpeak           slp             caa        
##  Min.   :0.0000   Min.   :0.000   Min.   :0.000   Min.   :0.0000  
##  1st Qu.:0.0000   1st Qu.:0.000   1st Qu.:1.000   1st Qu.:0.0000  
##  Median :0.0000   Median :0.800   Median :1.000   Median :0.0000  
##  Mean   :0.3278   Mean   :1.043   Mean   :1.397   Mean   :0.7185  
##  3rd Qu.:1.0000   3rd Qu.:1.600   3rd Qu.:2.000   3rd Qu.:1.0000  
##  Max.   :1.0000   Max.   :6.200   Max.   :2.000   Max.   :4.0000  
##      thall           output     
##  Min.   :0.000   Min.   :0.000  
##  1st Qu.:2.000   1st Qu.:0.000  
##  Median :2.000   Median :1.000  
##  Mean   :2.315   Mean   :0.543  
##  3rd Qu.:3.000   3rd Qu.:1.000  
##  Max.   :3.000   Max.   :1.000

\[~\] As we can see above:

• the people captured in this datasets represent people between 29 and 77 years, so is evident that there isn’t any teenager in this dataset. Because, there is a low probability that the teenagers have the heart attack in their young ages.
• the genders in this dataset are equally distributed between females and males.
• for the output features, we observed that there is again an equally distribution between people that have heart attack and not.

Other features of this dataset could be easily see in this summary, so let’s go to the data visualization for having a good visualizations of which data we are treating!

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The Data Visualization

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In this subsection we want to describe graphically the dataset that we are analyzing to have a first taste of what is the better model to use in this case, we want to show below some interesting plots:

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The visualization using PCA

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PCA is a particular tool that allows us to show the behaviour of the patients (treated as points in this space) of our data. We see in this way which are the patients more similar to each other

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Its normal to denote that there could be some losses on this representation due to the amount of features that we want to consider, but here we actually recover 89.8% of variance in the entire dataset using two principal components, so this is a good result.

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The Categorical Data

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Here, we want to analyze the percentages of each categorical data:

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The most relevant is the typical angina (47.4%) that is defined as substernal chest pain precipitated by physical exertion or emotional stress and relieved with rest or nitroglycerin (represents inadequate flow of blood and oxygen to the heart). Women and elderly patients are usually have atypical symptoms both at rest and during stress, often in the setting of nonobstructive coronary artery disease (CAD).

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the ST/heart rate slope (ST/HR slope), has been proposed as a more accurate ECG criterion for diagnosing significant coronary artery disease (CAD). The most relevant rate slope is approximately between the first and second type (as flat and as down sloping).

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The most case is about 0 major vessels that aren’t affect of any damaged, but in other little case we see that there are a bunch of major vessels that have a problem, within the heart.

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the maximum rate achieved is about a fixed defect.

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Angina may feel like pressure in the chest, jaw or arm. It frequently may occur with exercise or stress. Some people with angina also report feeling lightheaded, overly tired, short of breath or nauseated. As the heart pumps harder to keep up with what you are doing, it needs more oxygen-rich blood. This reflects to the fact that we have no exercise induced by angina.

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The Quantitative Data

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Here, we illustrate the main features of the quantitative data, after have seen the categorical data in the previous section:

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hist.and.summary('age', 'Persons Age')

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   29.00   48.00   55.50   54.42   61.00   77.00

hist.and.summary('chol', 'Cholestoral')

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   126.0   211.0   240.5   246.5   274.8   564.0

hist.and.summary('thalachh', 'Maximum Heart Rate Achieved')

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    71.0   133.2   152.5   149.6   166.0   202.0

hist.and.summary('trtbps', 'Resting Blood Pressure')

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    94.0   120.0   130.0   131.6   140.0   200.0

hist.and.summary('oldpeak', 'Previous Peak')

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   0.000   0.800   1.043   1.600   6.200

\[~\] As we can see above, the distribution of these data are different and several are more similar, for example if you compare the maximum heart rate and the age of the people these distributions are different! So, these considerations could be useful for predicting the heart attack.

After this, we consider also the densities considering the case of having the heart attack and not:

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dense.chart('age', 'Persons Age')

dense.chart('chol', 'Cholestoral')

dense.chart('thalachh', 'Maximum Heart Rate Achieved')

dense.chart('trtbps', 'Resting Blood Pressure')

dense.chart('oldpeak', 'Previous Peak')

The Correlations

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Here, we want to highlight the correlations between the features treated in qualitative, quantitative and without any distinction.

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With these correlation maps we see some interesting correlations is important to denote that the quantitative variables are more significant than the qualitative variables, because we are treating the quantitative measures, but here we can highlight and consider also the qualitative variables. So, here we denote in the complete map that the:

• thalachh and age \(\rightarrow\) have -0.40 of correlations
• cp and exng \(\rightarrow\) have -0.39 of correlations
• thalachh and exng \(\rightarrow\) have -0.38 of correlations
• slp and oldpeak \(\rightarrow\) have -0.58 of correlations

seems that these variables will be important in our predictions later, we will see in few moment if it is in this way!

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Preliminary brief definitions

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• Likelihood \(\pi(y_{obs}|\theta)\): measures the goodness of fit of a statistical model to a sample of data for giving values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.

• Logistic regression model: Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary). Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more nominal, ordinal, interval or ratio-level independent variables.

• logit link function: it uses the Cumulative Distribution Function (CDF) of the logistic distribution. A benefit to using Logit is that the coefficients can be interpreted in the terms of odds ratios.

• cloglog function: Is unlike Logit and Probit asymmetric. It is best used when the probability of an event is very small or very large. The complementary log-log approaches 0 infinitely slower than any other link function. Cloglog model is closely related to continuous-time models for the occurrence of events, so it has an important application in the area of survival analysis and hazard modeling.

• Prior \(\pi(\theta)\): is the probability distribution that would express one’s beliefs about this quantity before some evidence is taken into account.

• the deviance: is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing. It is a generalization of the idea of using the sum of squares of residuals (RSS) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood.

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The Goal

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The main goal is to leverage the main fully Bayesian analysis, based on understanding if a person has a heart attack or not.

So, the response variable \(Y_i\) (that is the “output” feature in our dataset) is the heart attack \(\in \{0,1\}\) and the predictor variables (\(x_i \in \mathbb{R}^{+}\)) have been chosen using the glm() function (we are cheating, but in this way we can sure that we are catching the right variables), used to fit generalized linear models, as we can see below:

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## 
## Call:
## glm(formula = y ~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + 
##     x10 + x11 + x12 + x13, family = binomial(link = "logit"), 
##     data = dat)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.5751  -0.3868   0.1514   0.5841   2.6239  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  3.304169   2.577891   1.282 0.199936    
## x1          -0.001469   0.023464  -0.063 0.950062    
## x2          -1.750930   0.468199  -3.740 0.000184 ***
## x3           0.847283   0.185548   4.566 4.96e-06 ***
## x4          -0.020188   0.010386  -1.944 0.051916 .  
## x5          -0.004489   0.003806  -1.179 0.238252    
## x6           0.073463   0.532452   0.138 0.890263    
## x7           0.450607   0.348505   1.293 0.196021    
## x8           0.023134   0.010449   2.214 0.026835 *  
## x9          -0.981017   0.409806  -2.394 0.016672 *  
## x10         -0.523604   0.214467  -2.441 0.014630 *  
## x11          0.589074   0.349864   1.684 0.092236 .  
## x12         -0.826015   0.201922  -4.091 4.30e-05 ***
## x13         -0.887203   0.290730  -3.052 0.002276 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 416.42  on 301  degrees of freedom
## Residual deviance: 210.35  on 288  degrees of freedom
## AIC: 238.35
## 
## Number of Fisher Scoring iterations: 6

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As we can see above, the glm rejects the hypothesis to consider the variables:

• Age ~ \(x_1\)
• Cholestoral ~ \(x_5\)
• fasting blood sugar ~ \(x_6\)
• resting electrocardiographic results ~ \(x_7\)

… and also the intercept isn’t a good choice to admit it in our model.

An important remark to highlight is that we tried to do the frequentist approach of these data and we should remember that considering all of the features we achieved 238.35 as AIC value.

But, if we try with the features that we decide to consider in the next steps?

summary(glm(y ~ x2+x3+x4+x8+x9+x10+x11+x12+x13, family = binomial(link = "logit"),data=dat))

## 
## Call:
## glm(formula = y ~ x2 + x3 + x4 + x8 + x9 + x10 + x11 + x12 + 
##     x13, family = binomial(link = "logit"), data = dat)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.5157  -0.3801   0.1652   0.5941   2.6269  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  2.554313   1.924575   1.327 0.184440    
## x2          -1.598915   0.433172  -3.691 0.000223 ***
## x3           0.843330   0.180454   4.673 2.96e-06 ***
## x4          -0.021336   0.009893  -2.157 0.031023 *  
## x8           0.022131   0.009334   2.371 0.017738 *  
## x9          -0.951980   0.401449  -2.371 0.017723 *  
## x10         -0.514275   0.208059  -2.472 0.013444 *  
## x11          0.611946   0.341449   1.792 0.073101 .  
## x12         -0.816831   0.194912  -4.191 2.78e-05 ***
## x13         -0.902166   0.279919  -3.223 0.001269 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 416.42  on 301  degrees of freedom
## Residual deviance: 214.19  on 292  degrees of freedom
## AIC: 234.19
## 
## Number of Fisher Scoring iterations: 6

The behaviour is slightly changed, we have 234.19 as AIC value, we will see if we achieve the same result!

Then, in summary we have N = 302 observations. Then we decided to consider two models and see the difference on which model at the end is the best model of our analysis.

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The First Model (link function with logit)

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In the first model, we decide to consider the following logistic regression model with the logit link function:

\[ Y_i \sim Bern(logit(p_i)) \\ logit(p_i) = \beta_{2} \cdot x_{2_i} + \beta_{3} \cdot x_{3_i} + \beta_{4} \cdot x_{4_i} +\beta_{8} \cdot x_{8_i} + \beta_{9} \cdot x_{9_i} + \beta_{10} \cdot x_{10_i} + \beta_{11} \cdot x_{11_i} + \beta_{12} \cdot x_{12_i} + \beta_{13} \cdot x_{13_i} \] The prior beta parameters are chosen approximatly considering the estimated averages saw in the previous subsection regarding the outcomes of glm(). So, the beta prior parameters are distributed following a normal distribution as follows:

\[ \beta_2 \sim N(\mu_2 = -1, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_3 \sim N(\mu_3 = 0.5, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_4 \sim N(\mu_4 = 0, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_8 \sim N(\mu_8 = 0, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_9 \sim N(\mu_9 = -0.5, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_{10} \sim N(\mu_{10} = 0.5, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_{11} \sim N(\mu_{11} = 0.5, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_{12} \sim N(\mu_{12} = -0.5, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \beta_{13} \sim N(\mu_{13} = -0.5, \tau^{2} = 1.0 \cdot 10^{-6}) \\ \]

The main reason is that we prefer chosing the binary classification as in this case the logistic regression to model our outcomes based only on 0 or 1.

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Implementing RJags

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As we can see, we implemented the model using RJags:

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• The model is defined as follows:

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model <- function(){
  # Likelihood
  for (i in 1:N){
    y[i] ~ dbern(p[i])
    logit(p[i]) <-  beta2*x2[i] + beta3*x3[i] + beta4*x4[i] + beta8*x8[i] + beta9*x9[i] + beta10*x10[i] + beta11*x11[i] + beta12*x12[i] + beta13*x13[i] # omit intercept and not useful variables
  }
  
  # Defining the prior beta parameters
  beta2 ~ dnorm(-1, 1.0E-6)
  beta3 ~ dnorm(0.5, 1.0E-6)
  beta4 ~ dnorm(0, 1.0E-6)
  beta8 ~ dnorm(0, 1.0E-6)
  beta9 ~ dnorm(-0.5, 1.0E-6)
  beta10 ~ dnorm(0.5, 1.0E-6)
  beta11 ~ dnorm(0.5, 1.0E-6)
  beta12 ~ dnorm(-0.5, 1.0E-6)
  beta13 ~ dnorm(-0.5, 1.0E-6)
}

remark: that the standard deviation in RJags is considered as the precision, so the \(\tau^2 = \frac{1}{\sigma^{2}}\)

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• Data structuring as requested with highlight which parameters to consider:

# Passing the data for RJags
data.jags <- list("y" = y, "N" = N,
                  "x2" = x2, "x3" = x3, "x4" = x4, "x8" = x8, "x9" = x9, "x10" = x10, "x11" = x11, "x12" = x12, "x13" = x13)

# Defining parameters of interest to show after running RJags
mod.params <- c("beta2", "beta3", "beta4", "beta8", "beta9", "beta10", "beta11", "beta12", "beta13")

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• Running the model with the following results:

# Run JAGS
set.seed(123)
n.chains <- 3
mod.fit <- jags(data = data.jags,                                                 # DATA
                model.file = model,                                               # MODEL
                parameters.to.save = mod.params,                                  # TRACKING
                n.chains = n.chains, n.iter = 10000, n.burnin = 1000, n.thin=10)  # MCMC

## module glm loaded

## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 302
##    Unobserved stochastic nodes: 9
##    Total graph size: 3840
## 
## Initializing model

mod.fit

## Inference for Bugs model at "C:/Users/Jeremy/AppData/Local/Temp/Rtmp4GRtjV/model233c3ab1e4e.txt", fit using jags,
##  3 chains, each with 10000 iterations (first 1000 discarded), n.thin = 10
##  n.sims = 2700 iterations saved
##          mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
## beta10    -0.489   0.212  -0.909  -0.628  -0.490  -0.344  -0.081 1.001  2700
## beta11     0.683   0.344   0.021   0.440   0.687   0.909   1.369 1.001  2700
## beta12    -0.846   0.199  -1.252  -0.977  -0.843  -0.711  -0.471 1.002  1900
## beta13    -0.851   0.277  -1.401  -1.038  -0.842  -0.659  -0.332 1.001  2700
## beta2     -1.652   0.445  -2.530  -1.955  -1.640  -1.356  -0.810 1.001  2700
## beta3      0.881   0.187   0.524   0.756   0.879   1.006   1.253 1.002  2700
## beta4     -0.015   0.008  -0.032  -0.020  -0.015  -0.009   0.001 1.002  1700
## beta8      0.031   0.008   0.016   0.026   0.031   0.037   0.047 1.002  1600
## beta9     -0.889   0.414  -1.688  -1.173  -0.895  -0.605  -0.069 1.001  2700
## deviance 225.361   4.463 218.784 222.067 224.697 227.789 235.802 1.001  2700
## 
## For each parameter, n.eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
## 
## DIC info (using the rule, pD = var(deviance)/2)
## pD = 10.0 and DIC = 235.3
## DIC is an estimate of expected predictive error (lower deviance is better).

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In this last running of the MCMC we have interesting results:

• mu.vect and sd.vect: these values representing the estimated values after simulating the MCMC.

• the percentages (quantiles): represents the quantiles of the interval where the parameter is likely in

• RHat: The values near 1 suggest convergence

• n.eff: is the effective number of samples to achieve the convergence, the stationary region.

just to have a look, let’s check how the fitting curve look like if we use the obtained estimates:

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The stationary regions

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Here, we show the univariate trace-plots of the simulations of each parameter:

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Some other interesting plots

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We are considering other interesting plots, underlining also each chain alone:

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Now, we want to see the autocorrelations of each parameter:

##               beta10       beta11       beta12       beta13       beta2
## Lag 0    1.000000000  1.000000000  1.000000000  1.000000000 1.000000000
## Lag 10   0.028749715  0.013027320  0.011401177  0.008061257 0.009605275
## Lag 50   0.004853036  0.007741027  0.006044363 -0.018532707 0.032820945
## Lag 100 -0.062010722 -0.017318088  0.005858652  0.022871691 0.044325748
## Lag 500  0.022963511  0.034443073 -0.006998113  0.004928576 0.002334635
##              beta3        beta4       beta8        beta9      deviance
## Lag 0   1.00000000  1.000000000  1.00000000  1.000000000  1.000000e+00
## Lag 10  0.02486498  0.001619878  0.04591646 -0.004177052 -1.524460e-02
## Lag 50  0.02580283 -0.048039671 -0.04290369 -0.023810367 -9.722008e-03
## Lag 100 0.02351930 -0.011874447 -0.01503457 -0.028663890 -9.939696e-05
## Lag 500 0.03144189  0.021556527  0.01383545  0.018783666  1.420195e-02

\[~\]

The ACFs follow good behaviours, because we need to have samples independent each other during new iterations. Here, we see that the correlations are going further from the first lag, this is strictly decreasing. So, this is a good point!

We can conferm our intuition also with the values returned by autocorr.diag() that it returned the vector of the average autocorrelations across all chains.

\[~\]

The Cumulative Means

\[~\] In this section we want to see the empirical average of \(\mathbf{\hat{I}}_{t}\) increasing the value of \(t \, = \, 1,...,T\). Before starting, we want to write the formula of \(\mathbf{\hat{I}}_{t}\) that is:

\[~\]

\[ \mathbf{I} \approx \mathbf{\hat{I}}_{t} = \frac{1}{T} \sum_{i=1}^{T} h(\theta^{(i)}) \]

\[~\]

is important to write \(\approx\) because this leverages the SLLN theorem and we want also to confirm this assumption! So, let’s see the results:

\[~\]

\[~\]

The Approximation Error

\[~\]

Here, we want to analyze the best approximation error for the MCMC sampling. We consider, essentially the square root of the MCSE.

The variance formula in the MCMC samplig is:

\[ \mathbf{V}[\hat{I}_{t}] = \frac{Var_{\pi}[h(X_{1})]}{t_{eff}} = \Big( 1 + 2 \sum_{k=1}^{\infty} \rho_{k}\Big)\frac{\sigma^{2}}{T} \]

where \(t_{eff}\) is the effective sample size (ESS). The idea is to have a sort of “exchange rate” between dependent and independent samples.

We can say, if we have 1,000 samples from a certain Markov chain are worth about as much as 70 independent samples because the MCMC samples are highly correlated.

Or if you have 1,000 samples from a different Markov chain are worth about as much as 400 independent samples because although the MCMC samples are dependent, they’re weakly correlated.

…More formal details on the slides.

As we can see the model 1 has these effective samples size for each parameters involved in this analysis:

##   beta10   beta11   beta12   beta13    beta2    beta3    beta4    beta8 
##     2700     2700     1900     2700     2700     2700     1700     1600 
##    beta9 deviance 
##     2700     2700

means that our samples, respect 10000 iterations, are weakly correlated.

…now let’s move on to calculate the MCSE

in this case we want to consider the MCSE that is the square root of the formula written above, So the results are (for each chain):

n <- length(colnames(mod.fit$BUGSoutput$sims.matrix))

mcse_dataframe <- data.frame(MCSE_jointly = rep(NA, n))

rownames(mcse_dataframe) <- colnames(mod.fit$BUGSoutput$sims.matrix)[1:n] 

for(colname in colnames(mod.fit$BUGSoutput$sims.matrix)[1:n]){
  mcse_dataframe[colname, "MCSE_jointly"] <- LaplacesDemon::MCSE(mod.fit$BUGSoutput$sims.matrix[ , colname])
}

mcse_dataframe

##          MCSE_jointly
## beta10   0.0039161885
## beta11   0.0068608122
## beta12   0.0038219783
## beta13   0.0053805117
## beta2    0.0084766703
## beta3    0.0035720520
## beta4    0.0001623317
## beta8    0.0001547387
## beta9    0.0082876402
## deviance 0.0920766126

\[~\]

As we can see the \(\beta_2\) has the highest uncertainty considering the jointly chains.

\[~\]

Posterior uncertainty

\[~\]

The uncertainty is measured by using the variability of the parameter w.r.t. its absolute expectation:

##                  mean          sd variability
## beta10    -0.48896601 0.211591852  0.43273325
## beta11     0.68316497 0.343508862  0.50281978
## beta12    -0.84554789 0.198878026  0.23520611
## beta13    -0.85142734 0.277368178  0.32576846
## beta2     -1.65152293 0.444823897  0.26934164
## beta3      0.88106225 0.186628248  0.21182186
## beta4     -0.01462894 0.008258434  0.56452715
## beta8      0.03139029 0.007968318  0.25384658
## beta9     -0.88885032 0.414351227  0.46616536
## deviance 225.36094747 4.463078811  0.01980414

The highest posterior uncertainty is about the \(\beta_4\).

\[~\]

The Estimated Correlations

\[~\]

Here, we bring the correlations between all the values calculated during the MCMC sampling, considering the joining matrix:

the highest correlation is between \(\beta_4 \text{ and } \beta_8\).

\[~\]

Hypotheses and Testing of the convergences of these multiple Markov chains

\[~\]

Here, we want to understand if we achieve with these multiple simulations of the markov chains the convergences and the validity. We are considering different tools saw in the last lectures, to understand if we achieve or not the convergency, the tools utilized are:

• The Raftery & Lewis Diagnostic
• The Geweke Diagnostic
• The Gelman Diagnostic
• The Heidel Diagnostic

\[~\]

Raftery & Lewis Test

\[~\] Introduced an MCMC diagnostic that estimates the number of iterations needed for a given level of precision in posterior samples, as well as estimating burn-in, when quantiles are the posterior summaries of interest.

coda::raftery.diag(coda.fit)

## [[1]]
## 
## Quantile (q) = 0.025
## Accuracy (r) = +/- 0.005
## Probability (s) = 0.95 
## 
## You need a sample size of at least 3746 with these values of q, r and s
## 
## [[2]]
## 
## Quantile (q) = 0.025
## Accuracy (r) = +/- 0.005
## Probability (s) = 0.95 
## 
## You need a sample size of at least 3746 with these values of q, r and s
## 
## [[3]]
## 
## Quantile (q) = 0.025
## Accuracy (r) = +/- 0.005
## Probability (s) = 0.95 
## 
## You need a sample size of at least 3746 with these values of q, r and s

\[~\]

Geweke Test

\[~\] This diagnostic is based on a test for equality of the means of the first and last part of a Markov chain (by default the first 10% and the last 50%). If the samples are drawn from a stationary distribution of the chain, then the two means are equal and Geweke’s statistic has an asymptotically standard normal distribution.

The test statistic is a standard Z-score: the difference between the two sample means divided by its estimated standard error. The standard error is estimated from the spectral density at zero, and so takes into account any autocorrelation.

The Z-score is calculated under the assumption that the two parts of the chain are asymptotically independent.

coda::geweke.diag(coda.fit)

## [[1]]
## 
## Fraction in 1st window = 0.1
## Fraction in 2nd window = 0.5 
## 
##   beta10   beta11   beta12   beta13    beta2    beta3    beta4    beta8 
##  -1.5996  -0.7659  -1.4946  -0.9632   1.0605   1.1048   0.9854  -0.0797 
##    beta9 deviance 
##  -1.2897   1.4878 
## 
## 
## [[2]]
## 
## Fraction in 1st window = 0.1
## Fraction in 2nd window = 0.5 
## 
##    beta10    beta11    beta12    beta13     beta2     beta3     beta4     beta8 
## -0.644081 -1.156059  0.004023  0.583586  0.554949  0.684522  1.291582 -1.392050 
##     beta9  deviance 
##  0.775653 -0.644099 
## 
## 
## [[3]]
## 
## Fraction in 1st window = 0.1
## Fraction in 2nd window = 0.5 
## 
##   beta10   beta11   beta12   beta13    beta2    beta3    beta4    beta8 
##   0.8445  -0.1983   1.0661   1.7963  -0.8795   1.4666  -1.1209  -0.3697 
##    beta9 deviance 
##   0.8451  -0.8643

\[~\]

Gelman Test

\[~\]

The `potential scale reduction factor’ is calculated for each variable in x, together with upper and lower confidence limits. Approximate convergence is diagnosed when the upper limit is close to 1. For multivariate chains, a multivariate value is calculated that bounds above the potential scale reduction factor for any linear combination of the (possibly transformed) variables.

The confidence limits are based on the assumption that the stationary distribution of the variable under examination is normal. Hence the `transform’ parameter may be used to improve the normal approximation.

coda::gelman.diag(coda.fit)

## Potential scale reduction factors:
## 
##          Point est. Upper C.I.
## beta10        1.000       1.00
## beta11        1.000       1.00
## beta12        1.000       1.00
## beta13        0.999       1.00
## beta2         1.001       1.00
## beta3         1.001       1.00
## beta4         1.002       1.01
## beta8         1.002       1.00
## beta9         1.000       1.00
## deviance      1.003       1.01
## 
## Multivariate psrf
## 
## 1

\[~\]

Heidel Test

\[~\] Implements a convergence diagnostic, and removes up to half the chain in order to ensure that the means are estimated from a chain that has converged.

The convergence test uses the Cramer-von-Mises statistic to test the null hypothesis that the sampled values come from a stationary distribution.

The test is successively applied, firstly to the whole chain, then after discarding the first 10%, 20%, … of the chain until either the null hypothesis is accepted, or 50% of the chain has been discarded.

The latter outcome constitutes `failure’ of the stationarity test and indicates that a longer MCMC run is needed. If the stationarity test is passed, the number of iterations to keep and the number to discard are reported.

The half-width test calculates a 95% confidence interval for the mean, using the portion of the chain which passed the stationarity test.

Half the width of this interval is compared with the estimate of the mean. If the ratio between the half-width and the mean is lower than eps, the halfwidth test is passed. Otherwise the length of the sample is deemed not long enough to estimate the mean with sufficient accuracy.

\[~\]

## [[1]]
##                                        
##          Stationarity start     p-value
##          test         iteration        
## beta10   passed       1         0.3239 
## beta11   passed       1         0.6880 
## beta12   passed       1         0.2287 
## beta13   passed       1         0.7323 
## beta2    passed       1         0.7564 
## beta3    passed       1         0.7176 
## beta4    passed       1         0.0622 
## beta8    passed       1         0.2670 
## beta9    passed       1         0.6451 
## deviance passed       1         0.1120 
##                                      
##          Halfwidth Mean     Halfwidth
##          test                        
## beta10   passed     -0.4835 0.014111 
## beta11   passed      0.6757 0.020904 
## beta12   passed     -0.8545 0.012844 
## beta13   passed     -0.8541 0.017958 
## beta2    passed     -1.6552 0.029211 
## beta3    passed      0.8795 0.012322 
## beta4    passed     -0.0147 0.000559 
## beta8    passed      0.0316 0.000491 
## beta9    passed     -0.8927 0.025430 
## deviance passed    225.2703 0.297930 
## 
## [[2]]
##                                        
##          Stationarity start     p-value
##          test         iteration        
## beta10   passed       1         0.233  
## beta11   passed       1         0.254  
## beta12   passed       1         0.855  
## beta13   passed       1         0.784  
## beta2    passed       1         0.699  
## beta3    passed       1         0.153  
## beta4    passed       1         0.642  
## beta8    passed       1         0.812  
## beta9    passed       1         0.423  
## deviance passed       1         0.769  
##                                      
##          Halfwidth Mean     Halfwidth
##          test                        
## beta10   passed     -0.4928 0.013672 
## beta11   passed      0.6839 0.022893 
## beta12   passed     -0.8391 0.013463 
## beta13   passed     -0.8527 0.016625 
## beta2    passed     -1.6389 0.028610 
## beta3    passed      0.8842 0.011386 
## beta4    passed     -0.0149 0.000536 
## beta8    passed      0.0315 0.000566 
## beta9    passed     -0.8891 0.027043 
## deviance passed    225.3264 0.268161 
## 
## [[3]]
##                                        
##          Stationarity start     p-value
##          test         iteration        
## beta10   passed       1         0.3448 
## beta11   passed       1         0.1388 
## beta12   passed       1         0.6863 
## beta13   passed       1         0.1368 
## beta2    passed       1         0.0714 
## beta3    passed       1         0.3191 
## beta4    passed       1         0.5506 
## beta8    passed       1         0.8238 
## beta9    passed       1         0.8632 
## deviance passed       1         0.5795 
##                                      
##          Halfwidth Mean     Halfwidth
##          test                        
## beta10   passed     -0.4906 0.013692 
## beta11   passed      0.6899 0.022293 
## beta12   passed     -0.8431 0.012653 
## beta13   passed     -0.8475 0.018361 
## beta2    passed     -1.6604 0.031817 
## beta3    passed      0.8795 0.014677 
## beta4    passed     -0.0142 0.000546 
## beta8    passed      0.0310 0.000524 
## beta9    passed     -0.8847 0.031369 
## deviance passed    225.4861 0.294261

\[~\]

We prefer considering the Heidel test because it is a double check of the stationarity and the convergency states. This is sometimes failed, meanwhile the other tools seems to be good, so I prefer considering this test.

Here, we can see all is considered correctly, so our intuitions are correct!

\[~\]

Credible interval and the point estimates

\[~\]

After checking that the simulations returned to pass the convergence, we can decide to pool up all the chains and does the credible interval and the point estimates for our beta values estimated:

chainMat <- mod.fit$BUGSoutput$sims.matrix

# Point estimates
(beta.hat.jags <- colMeans(chainMat))

##       beta10       beta11       beta12       beta13        beta2        beta3 
##  -0.48896601   0.68316497  -0.84554789  -0.85142734  -1.65152293   0.88106225 
##        beta4        beta8        beta9     deviance 
##  -0.01462894   0.03139029  -0.88885032 225.36094747

# Credible Intervals
cred <- 0.95
(p.ET.jags <- apply(chainMat, 2, quantile, 
                    prob=c((1-cred)/2, 1-(1-cred)/2)))

##           beta10     beta11     beta12     beta13      beta2     beta3
## 2.5%  -0.9087009 0.02076826 -1.2517219 -1.4006366 -2.5300359 0.5235355
## 97.5% -0.0813963 1.36922157 -0.4707176 -0.3323277 -0.8100347 1.2534855
##              beta4      beta8       beta9 deviance
## 2.5%  -0.031580570 0.01601970 -1.68838742 218.7839
## 97.5%  0.001399235 0.04741742 -0.06901415 235.8023

# What about the HPD?
(p.HPD.jags <- coda::HPDinterval(as.mcmc(chainMat)))

##                 lower         upper
## beta10    -0.89921219  -0.076049113
## beta11     0.04745225   1.391044537
## beta12    -1.23004141  -0.459179061
## beta13    -1.40111307  -0.332202228
## beta2     -2.53140813  -0.809784615
## beta3      0.51654945   1.243256608
## beta4     -0.03264154   0.000171662
## beta8      0.01489135   0.046198108
## beta9     -1.65197093  -0.037787699
## deviance 218.22568870 234.064380517
## attr(,"Probability")
## [1] 0.95

It’s relevant to conclude that all your parameters (and its variables) are significant to predict heart attacks (except for \(\beta_4\))

\[~\]

The Predictions

\[~\]

Here, we want to illustrate how using the markov chain to approximate the posterior predictive distribution of the people that could have the heart attack or not considering the features explained above. So, we want to approximate this area:

\[ m(y_{new}) = \int_{\Theta} f(y_{new}|\theta, Y_{old}, x_2, x_3, x_4, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13})\pi(\theta)d\theta \]

where the \(\theta\) is equal to the coefficients used and considered before, \(\Theta\) is the parameter space and \(\pi(\theta)\) is the product of the priors thanks to marginal independence assumption. At the end, we want to smaple from the \(\pi(\theta)\) and from \(f(y_{new}|\theta, Y_{old}, x_2, x_3, x_4, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13})\).

Here, there is a brief practical example:

new_record <- list(x2 = sample(x2, 1, replace = T), x3 = sample(x3, 1, replace = T), x4 = sample(x4, 1, replace = T), x8 = sample(x8, 1, replace = T), x9 = sample(x9, 1, replace = T),
                   x10 = sample(x10, 1, replace = T), x11 = sample(x11, 1, replace = T), x12 = sample(x12, 1, replace = T), x13 = sample(x13, 1, replace = T))

x <- mod.fit$BUGSoutput$sims.matrix[,"beta2"]*new_record$x2 + mod.fit$BUGSoutput$sims.matrix[,"beta3"]*new_record$x3 + mod.fit$BUGSoutput$sims.matrix[,"beta4"]*new_record$x4 + mod.fit$BUGSoutput$sims.matrix[,"beta8"]*new_record$x8 + mod.fit$BUGSoutput$sims.matrix[,"beta9"]*new_record$x9 + mod.fit$BUGSoutput$sims.matrix[,"beta10"]*new_record$x10 + mod.fit$BUGSoutput$sims.matrix[,"beta11"]*new_record$x11 + mod.fit$BUGSoutput$sims.matrix[,"beta12"]*new_record$x12 + mod.fit$BUGSoutput$sims.matrix[,"beta13"]*new_record$x13 
x_mu <- 1/(1+exp(-x))

y_pred <- unlist(lapply(x_mu, function(x) rbinom(n = 1, size = 1, prob = x)))
sd(y_pred)

## [1] 0.3478183

prop.table(table(y_pred))

## y_pred
##         0         1 
## 0.8592593 0.1407407

prop.table(table(y))

## y
##         0         1 
## 0.4569536 0.5430464

summary(y_pred)[4]

##      Mean 
## 0.1407407

hchart(y_pred, type = "column", name = "model1", color = randomcoloR::randomColor()) %>%
  hc_title(text = "Predictions with model 1") %>%
  hc_xAxis(title = "model1") %>%
  hc_chart(options3d=list(enabled=TRUE, alpha=2, beta=-10, 
                          depth=100, viewDistance=25)) %>% 
  hc_plotOptions(column=list(depth= 100))

After read, the whole project report we will see which is the best model here in terms of predictions, stay tuned!

\[~\]

The Second Model

\[~\] Now, we want to focus to another model always the logistic regression with the whole features and a new link function, the cloglog function.

Why this changment? Because, we want to see which model is better… and how can you see this? I’ll tell you where you can easy see this better preference.

What is the link function? So, the link function transforms the probabilities of the levels of a categorical response variable to a continuous scale that is unbounded. Once the transformation is complete, the relationship between the predictors and the response can be modeled with the logistic regression for example.

As we can see we will reproduce this second model changing the link function to see if it is better or not than the previous model:

\[~\]

## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 302
##    Unobserved stochastic nodes: 13
##    Total graph size: 5499
## 
## Initializing model

## Inference for Bugs model at "C:/Users/Jeremy/AppData/Local/Temp/Rtmp4GRtjV/model233c4ac84acc.txt", fit using jags,
##  3 chains, each with 10000 iterations (first 1000 discarded), n.thin = 10
##  n.sims = 2700 iterations saved
##          mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
## beta1      0.000   0.000   0.000   0.000   0.000   0.000   0.000 1.003  1100
## beta10    -0.368   0.131  -0.634  -0.455  -0.367  -0.278  -0.123 1.001  2400
## beta11     0.233   0.207  -0.177   0.091   0.231   0.370   0.642 1.003   700
## beta12    -0.019   0.013  -0.046  -0.028  -0.019  -0.010   0.006 1.002  1800
## beta13    -0.563   0.168  -0.897  -0.673  -0.560  -0.453  -0.231 1.002  1100
## beta2     -1.135   0.249  -1.638  -1.310  -1.131  -0.961  -0.658 1.001  2700
## beta3      0.528   0.108   0.314   0.456   0.526   0.599   0.744 1.002  1500
## beta4      0.000   0.000   0.000   0.000   0.000   0.000   0.000 1.000     1
## beta5      0.000   0.000   0.000   0.000   0.000   0.000   0.000 1.000     1
## beta6     -0.044   0.289  -0.631  -0.238  -0.041   0.147   0.513 1.001  2700
## beta7      0.347   0.199  -0.043   0.215   0.349   0.479   0.742 1.001  2700
## beta8      0.017   0.004   0.010   0.014   0.017   0.019   0.024 1.007   750
## beta9     -0.594   0.276  -1.147  -0.783  -0.587  -0.412  -0.067 1.001  2500
## deviance 246.342   5.059 238.164 242.551 245.727 249.433 258.247 1.000  2700
## 
## For each parameter, n.eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
## 
## DIC info (using the rule, pD = var(deviance)/2)
## pD = 12.8 and DIC = 259.1
## DIC is an estimate of expected predictive error (lower deviance is better).

\[~\]

Can you see something different? Seems yes.. see the DIC! Is better the previous model (although the difference is too tiny, but the previous model is better).

…mmh, what is the DIC?

\[~\]

The comparison between these two models (DIC)

\[~\]

As we can see above, we tried to make a different model only changing the link function to see if it is better to associate with our data, the DIC (Deviance information criterion) is our indicator that says to us which is the better model.

The deviance information criterion (DIC) is a hierarchical modeling generalization of the Akaike information criterion (AIC). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation. DIC is an asymptotic approximation as the sample size becomes large, like AIC.

the DIC is calculated as:

\[ DIC = p_D + \overline{D(\theta)} \]

where:

• there are two calculations in common usage for the effective number of parameters of the model.
  • the first, as described in Spiegalhalter et al. (2002, p.587) is p_D = \(\overline{D(\theta)} - D(\overline{\theta})\), where \(\overline{\theta}\) is the expectation of \(\theta\)
  • …and the second described in Gelman et al. (2004, p.587) is \(p_D = \frac{1}{2}\mathbb{V}(D(\theta))\).

The larger the effective number of parameters is, the easier it is for the model to fit the data, and so the deviance needs to be penalized.

• the deviance is \(D(\theta) = -2log(p(y|\theta)) + C\), where y are the data, \(\theta\) are unknown parameters of the model and C is a constant.

Lower is the DIC value better is the accuracy of the model, in this case is better the first model.

\[~\]

The Conclusions

\[~\] We try to plot the logistic regression using the parameters obtained in the first model, we compare the line for each parameter-response variable:

We prefer considering to show only the significant ones.

\[~\]

Simulation for parameters recovery

\[~\] Now, we understood that the first model is better than the second. So, for checking that the model proposed can correctly recover the model parameters. I Executed the simulation considering the data simulated from the model proposed, the beta parameters checked are the estimated from the first model:

• \(\beta_2 \rightarrow -1.65\)
• \(\beta_3 \rightarrow 0.88\)
• \(\beta_4 \rightarrow -0.01\)
• \(\beta_8 \rightarrow 0.03\)
• \(\beta_9 \rightarrow -0.88\)
• \(\beta_{10} \rightarrow -0.48\)
• \(\beta_{11} \rightarrow 0.68\)
• \(\beta_{12} \rightarrow -0.84\)
• \(\beta_{13} \rightarrow -0.85\)

## module glm loaded

## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 302
##    Unobserved stochastic nodes: 9
##    Total graph size: 3890
## 
## Initializing model

## Inference for Bugs model at "C:/Users/Jeremy/AppData/Local/Temp/RtmpOexlNl/model5bf4139040e9.txt", fit using jags,
##  3 chains, each with 10000 iterations (first 1000 discarded), n.thin = 10
##  n.sims = 2700 iterations saved
##          mu.vect sd.vect    2.5%     25%     50%     75%   97.5%  Rhat n.eff
## beta10    -0.417   0.134  -0.683  -0.508  -0.412  -0.324  -0.162 1.001  2700
## beta11     0.221   0.248  -0.252   0.053   0.221   0.389   0.706 1.003   880
## beta12    -0.007   0.267  -0.549  -0.184  -0.009   0.168   0.531 1.000  2700
## beta13    -0.796   0.252  -1.281  -0.960  -0.798  -0.631  -0.314 1.001  2700
## beta2      2.092   0.331   1.444   1.870   2.090   2.318   2.731 1.002  1700
## beta3      0.719   0.239   0.247   0.563   0.720   0.876   1.191 1.000  2700
## beta4     -0.018   0.006  -0.030  -0.022  -0.017  -0.013  -0.006 1.002  1600
## beta8      0.004   0.005  -0.006   0.000   0.004   0.007   0.014 1.002  1600
## beta9      1.213   0.437   0.355   0.929   1.211   1.493   2.098 1.001  2700
## deviance 308.850   4.467 302.412 305.585 308.098 311.195 319.392 1.002  1200
## 
## For each parameter, n.eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
## 
## DIC info (using the rule, pD = var(deviance)/2)
## pD = 10.0 and DIC = 318.8
## DIC is an estimate of expected predictive error (lower deviance is better).

The values are slighlty the same, so the parameters chose are perfectly good!

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… Final conclusions

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We saw the power and the weakness of Bayesian Analysis.

At first step we tried to face with the model on jags “cheating” a little bit on which parameters are maybe good for our purposes and after that we make our usual considerations, then we compared with another model not so much good, Of curse many little things could be improved in the future in this amazing analysis that I did in my opinion.

So thanks for all!

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The References

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1. Heart Attack Analysis & Prediction Dataset - [Kaggle] https://www.kaggle.com/rashikrahmanpritom/heart-attack-analysis-prediction-dataset

2. For graphical plots - [Kaggle] https://www.kaggle.com/chaitanya99/heart-attack-analysis-prediction/data

3. Link functions, the model types - [Alteryx] https://community.alteryx.com/t5/Alteryx-Designer-Knowledge-Base/Selecting-a-Logistic-Regression-Model-Type-Logit-Probit-or/ta-p/111269

4. HighCharter, for the amazing plots in R - [High Charter] https://jkunst.com/highcharter/articles/hchart.html

5. Coronary artery diasease - [PubMed] https://pubmed.ncbi.nlm.nih.gov/3739881

6. Nitroglycerin - [Wikipedia] https://en.wikipedia.org/wiki/Nitroglycerin

7. Vessel Diases - [Digirad] https://www.digirad.com/triple-vessel-disease-diagnosis-treatment/

8. Different types of Angina - [CardioSmart] https://www.cardiosmart.org/topics/angina

9. Logistic Regression Bayesian Model R - [BayesBall] https://bayesball.github.io/BOOK/bayesian-multiple-regression-and-logistic-models.html

10. Typical Angina - [NCBI] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5680106

11. RDocumentations - [RDocumention] https://www.rdocumentation.org